Difference between revisions of "2005 AMC 10B Problems/Problem 23"
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Dividing the second equation by <math>z</math>, we get that <math>2(\tfrac{x}{z})=1+3(\tfrac{y}{z})</math>. | Dividing the second equation by <math>z</math>, we get that <math>2(\tfrac{x}{z})=1+3(\tfrac{y}{z})</math>. | ||
− | Now, when we subtract the top equation from the bottom, we get that <math>\tfrac{x}{z}= | + | Now, when we subtract the top equation from the bottom, we get that <math>\tfrac{x}{z}=\boxed{\textbf{(C) }5}</math> |
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==Solution 3== | ==Solution 3== |
Revision as of 15:43, 16 December 2021
Contents
[hide]Problem
In trapezoid we have parallel to , as the midpoint of , and as the midpoint of . The area of is twice the area of . What is ?
Solution 1
Since the heights of both trapezoids are equal, and the area of is twice the area of ,
.
, so
.
is exactly halfway between and , so .
, so
, and
.
.
Solution 2
Mark , , and Note that the heights of trapezoids & are the same. Mark the height to be .
Then, we have that .
From this, we get that .
We also get that .
Simplifying, we get that
Notice that we want .
Dividing the first equation by , we get that .
Dividing the second equation by , we get that .
Now, when we subtract the top equation from the bottom, we get that
Solution 3
Since the bases of the trapezoids along with the height are the same, the only thing that matters is the second base. Denote the length of the bigger trapezoid . The area of the smaller trapezoid is = . The area of the larger trapezoid is = . Since this problem asks for proportions, assume that and .
The smaller trapezoid has area while the larger trapezoid must have area . We have the equation . = 10, and our answer is
~Arcticturn
See Also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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